A problem of Cameron and Erdős, who proved that $f_m(n)>2^{n/4}$, and also asked whether\[f_m(n)=o(f(n)),\]where $f(n)$ counts the number of all (not necessarily maximal) sum-free sets. Luczak and Schoen [LuSc01] proved that there exists a constant $c<1/2$ such that\[f_m(n)<2^{cn},\]resolving these questions. Balogh, Liu, Sharifzadeh, and Treglown [BLST15] proved that\[f_m(n)=2^{(\frac{1}{4}+o(1))n},\]which the same authors [BLST18] later improved to\[f_m(n)=(C_n+o(1))2^{n/4},\]where $C_n$ is some explicit constant depending only on $n\pmod{4}$.

See [748] for the non-maximal case.

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This page was last edited 02 December 2025.